User:Tohline/SR/IdealGas

Ideal Gas Equation of State

Much of the following overview of ideal gas relations is drawn from Chapter II of Chandrasekhar's classic text on Stellar Structure [C67], which was originally published in 1939. A guide to parallel print media discussions of this topic is provided alongside the ideal gas equation of state in the key equations appendix of this H_Book.

Fundamental Properties of an Ideal Gas

Property #1

An ideal gas containing <math>~n_g</math> free particles per unit volume will exert on its surroundings an isotropic pressure (i.e., a force per unit area) <math>~P</math> given by the following

Standard Form
of the Ideal Gas Equation of State,

<math>~P = n_g k T</math>

[C67], Chapter VII.3, Eq. (18)
D. D. Clayton (1968), Eq. (2-7)
[H87], §1.1, p. 5

if the gas is in thermal equilibrium at a temperature <math>~T</math>.

Property #2

The internal energy per unit mass <math>~\epsilon</math> of an ideal gas is a function only of the gas temperature <math>~T</math>, that is,

<math>~\epsilon = \epsilon(T) \, .</math>

[C67], Chapter II, Eq. (1)

Specific Heats

Drawing from Chapter II, §1 of [C67]:  "Let <math>~\alpha</math> be a function of the physical variables. Then the specific heat, <math>~c_\alpha</math>, at constant <math>~\alpha</math> is defined by the expression,"

<math>~c_\alpha</math>

<math>~\equiv</math>

<math>~\biggl( \frac{dQ}{dT} \biggr)_{\alpha ~=~ \mathrm{constant}}</math>

The specific heat at constant pressure <math>~c_P</math> and the specific heat at constant (specific) volume <math>~c_V</math> prove to be particularly interesting parameters because they identify experimentally measurable properties of a gas.

From the Fundamental Law of Thermodynamics, namely,

<math>~dQ</math>

<math>~=</math>

<math>~ d\epsilon + PdV \, , </math>

it is clear that when the state of a gas undergoes a change at constant (specific) volume <math>~(dV = 0)</math>,

<math>~\biggl( \frac{dQ}{dT} \biggr)_{V ~=~ \mathrm{constant}}</math>	<math>~=</math>	<math>~\frac{d\epsilon}{dT}</math>
<math>~\Rightarrow ~~~ c_V</math>	<math>~=</math>	<math>~\frac{d\epsilon}{dT} \, .</math>

Assuming <math>~c_V</math> is independent of <math>~T</math> — a consequence of the kinetic theory of gasses; see, for example, Chapter X of [C67] — and knowing that the specific internal energy is only a function of the gas temperature — see Property #2 above — we deduce that,

<math>~\epsilon</math>

<math>~=</math>

<math>~c_V T \, .</math>

[C67], Chapter II, Eq. (10)
[LL75], Chapter IX, §80, Eq. (80.10)
[H87], §1.2, p. 9
[HK94], §3.7.1, immediately following Eq. (3.80)

Also, from Form A of the Ideal Gas Equation of State (see below) and the recognition that <math>~\rho = 1/V</math>, we can write,

<math>~P_\mathrm{gas}V</math>	<math>~=</math>	<math>~\biggl(\frac{\Re}{\bar\mu} \biggr) T</math>
<math>~\Rightarrow ~~~ PdV + VdP</math>	<math>~=</math>	<math>~\biggl(\frac{\Re}{\bar\mu} \biggr) dT \, .</math>

As a result, the Fundamental Law of Thermodynamics can be rewritten as,

<math>~dQ</math>

<math>~=</math>

<math>~c_\mathrm{V} dT + \biggl(\frac{\Re}{\bar\mu} \biggr) dT - VdP \, .</math>

This means that the specific heat at constant pressure is given by the relation,

<math>~c_P \equiv \biggl( \frac{dQ}{dT} \biggr)_{P ~=~ \mathrm{constant}}</math>

<math>~=</math>

<math>~c_V + \frac{\Re}{\bar\mu} \, .</math>

That is,

<math>~c_P - c_V </math>

<math>~=</math>

<math>~\frac{\Re}{\bar\mu} \, .</math>

[C67], Chapter II, §1, Eq. (9)
D. D. Clayton (1968), Eq. (2-108)
[LL75], Chapter IX, §80, immediately following Eq. (80.11)
[H87], §1.2, p. 9
[KW94], §4.1, immediately following Eq. (4.15)

Consequential Ideal Gas Relations

Throughout most of this H_Book, we will define the relative degree of compression of a gas in terms of its mass density <math>~\rho</math> rather than in terms of its number density <math>~n_g</math>. Following D. D. Clayton (1968) — see his p. 82 discussion of The Perfect Monatomic Nondegenerate Gas — we will "let the mean molecular weight of the perfect gas be designated by <math>~\bar{\mu}</math>. Then the density is

<math>~\rho = n_g \bar\mu m_u \, ,</math>

where <math>~m_u</math> is the mass of 1 amu" (atomic mass unit). "The number of particles per unit volume can then be expressed in terms of the density and the mean molecular weight as

<math>~n_g = \frac{\rho}{\bar\mu m_u} = \frac{\rho N_A}{\bar\mu} \, ,</math>

where <math>~N_A</math> = 1/<math>~m_u</math> is Avogadro's number …" Substitution into the above-defined Standard Form of the Ideal Gas Equation of State gives, what we will refer to as,

Form A
of the Ideal Gas Equation of State,

<math>~P_\mathrm{gas} = \frac{\Re}{\bar{\mu}} \rho T</math>

[LL75], Chapter IX, §80, Eq. (80.8)
[KW94], §2.2, Eq. (2.7) and §13, Eq. (13.1)

where <math>~\Re</math> ≡ <math>~k</math><math>~N_A</math> is generally referred to in the astrophysics literature as the gas constant. The definition of the gas constant can be found in the Variables Appendix of this H_Book; its numerical value can be obtained by simply scrolling the computer mouse over its symbol in the text of this paragraph. See §VII.3 (p. 254) of [C67] or §13.1 (p. 102) of [KW94] for particularly clear explanations of how to calculate <math>~\bar{\mu}</math>.

Employing a couple of the expressions from the above discussion of specific heats, the right-hand side of Form A of the Ideal Gas Equation of State can be rewritten as,

<math>~\frac{\Re}{\bar\mu} \rho T</math>

<math>~=</math>

<math>~ (c_P - c_V)\rho \biggl(\frac{\epsilon}{c_V}\biggr) = (\gamma_g - 1)\rho\epsilon \, , </math>

where we have — as have many before us — introduced a key physical parameter,

<math>~\gamma_g</math>

<math>~\equiv</math>

<math>~\frac{c_P}{c_V} \, ,</math>

[C67], Chapter II, immediately following Eq. (9)
[LL75], Chapter IX, §80, immediately following Eq. (80.9)
[T78], §3.4, immediately following Eq. (72)
[HK94], §3.7.1, Eq. (3.86)

to quantify the ratio of specific heats. This leads to what we will refer to as,

Form B
of the Ideal Gas Equation of State

<math>~P = (\gamma_\mathrm{g} - 1)\epsilon \rho </math>

[C67], Chapter II, Eq. (5)
[HK94], §1.3.1, Eq. (1.22)
[BLRY07], §6.1.1, Eq. (6.4)

Related Wikipedia Discussions

• Equation of State: Classical ideal gas law
• Ideal Gas Law
• Ideal Gas

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